Since we are interested in \(\sum _{i=1}^{r+1} a_T^{(r,m)}(i)\) , we are interested in the number of sequences of \(m\)  1s, \(m\)  2s, and so on up to \(m\)  \((r+1)\) s such that no two adjacent elements of the sequence are equal and the last element of the sequence is not equal to the first. As stated in [1]}, these are known as \(\emph {circular Smirnov words}\) . In general, these are cumbersome to enumerate, although there is a known generating function as given in [2]} . We recall the generating function of Smirnov words over alphabet \(\lbrace x_1,x_2,\ldots ,x_n\rbrace \)  is
\(S = \frac{1}{1-\sum _{i=1}^{n} \frac{x_i}{1+x_i}}\) 
